# Chain rule (multivariable calculus)

• Poetria
In summary, the conversation discusses the evaluation and approximation of functions ##f_x=3*x^2+y## and ##f_y=2*y+x##, as well as the calculation of the gradient and tangent at a given point. The participants use metaphors and references to mathematicians Leibniz and Émilie du Châtelet to describe the process of finding the linear approximation and tangent plane of a function at a specific point along its path.
Poetria
Homework Statement
Let ##f(x,y)=x^3+y^2+x*y##
Suppose that a point is moving through the plane. At time t , the point is at ## (x(t), y(t))=(t^2, e^{t-1})##. Use linear approximation to estimate the change in f as t goes from 1 to 1.1 . In other words, approximate
Relevant Equations
Multivariable chain rule
##f_x=3*x^2+y##
##f_y=2*y+x##

##(3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1}##
Well, I am not sure how to evaluate it.
I got a wrong result by multiplying by 0.1, i.e.
##((3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1})*0.1##

I guess it is trivial but I am lost. :(

We are interested in the change along ##t##. So why don't you write ##f(x,y)=f(x(t),y(t))=f(t)## in the first place? Then we have two points at ##t=1## and ##t=1.1##. The linear approximation is the secant through these points:
$$y=\underbrace{\dfrac{f(1)-f(1.1)}{1-1.1}}_{=:m}\cdot x + b\;\wedge\;f(1)=m\cdot 1+b$$
Finally, you can check the quality of this approximation by calculating the tangent at ##t=1##:
$$y=\left. \dfrac{d}{dt}\right|_{t=1}f(t) \cdot x + c =f'(1)x+c\;\wedge\;f(1)=f'(1)\cdot 1 +c$$
which is the linear approximation if only one point is given.

Poetria
Many thanks. But I can't read this:
"The linear approximation is the secant through these points:
You can't use 'macro parameter character #' in math mode ?

A nice quote from Leibniz. :) Émilie du Châtelet (my avatar) was Leibnizian. :)

Poetria said:
Many thanks. But I can't read this
You need to give me a second to correct my format. Please reload.

"He who hasn't tasted bitter things hasn't earned sweet things." (G. Leibniz)

"Let us choose for ourselves our path in life, and let us try to strew that path with flowers." (E. du Châtelet)

##t\longmapsto f(t)## is the path, and you should plant the flowers at ##t=1## and ##t=1.1.##

Poetria
fresh_42 said:
You need to give me a second to correct my format. Please reload.

"He who hasn't tasted bitter things hasn't earned sweet things." (G. Leibniz)

"Let us choose for ourselves our path in life, and let us try to strew that path with flowers." (E. du Châtelet)

##t\longmapsto f(t)## is the path, and you should plant the flowers at ##t=1## and ##t=1.1.##
I got:
-10.3022 + 13.3022*x = y
x=1.1
Approximation of the change:
4.33022-3=1.33022

A beautiful metaphor: ##t\longmapsto f(t)## is the path, and you should plant the flowers at ##t=1## and ##t=1.1.##

Perhaps a gradient would be also a good idea:
Gradient ##\vec (0.21, 0.105171)##
A tangent plane: 0.21*x + 0.105171*y-0.32

Slope of the gradient: 0.500814

Last edited:
Poetria said:
A beautiful metaphor: ##t\longmapsto f(t)## is the path, and you should plant the flowers at ##t=1## and ##t=1.1.##
Yes, but a little bit too fast posted. The path is actually ##t\longmapsto (x(t),y(t))## and ##f(t)## the function values along the path.

Poetria
fresh_42

## 1. What is the chain rule in multivariable calculus?

The chain rule is a fundamental concept in multivariable calculus that allows us to calculate the derivative of a composite function. In other words, it helps us find the rate of change of a function that depends on multiple variables.

## 2. How is the chain rule used in real-world applications?

The chain rule is used in a variety of real-world applications, such as physics, economics, and engineering. For example, it can be used to calculate the acceleration of a particle moving in multiple dimensions, or to optimize production processes in a factory.

## 3. What is the formula for the chain rule?

The formula for the chain rule is (f ◦ g)'(x) = f'(g(x)) * g'(x), where f and g are functions, f'(x) is the derivative of f with respect to x, and g'(x) is the derivative of g with respect to x.

## 4. Can the chain rule be extended to more than two functions?

Yes, the chain rule can be extended to any number of functions in a composition. For example, if we have a function h(x) = f(g(x)), we can use the chain rule to find h'(x) by taking the derivative of f with respect to g and multiplying it by the derivative of g with respect to x.

## 5. How does the chain rule relate to the product rule and quotient rule?

The chain rule is closely related to the product rule and quotient rule, as all three are methods for finding the derivative of composite functions. The product rule is used when two functions are multiplied together, while the quotient rule is used when one function is divided by another. The chain rule is used when one function is nested inside another.

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